1. Field of Invention
The invention is directed to the quantitative determination of sediment physical properties of the seabed and the layered subbottom using single-channel seismic-reflection profiling data.
2. Discussion of Related Art
The seismic-reflection profiling method using normal-incidence single-channel acoustic systems is widely used in marine surveys to provide a qualitative overview of the large-scale geological features within the upper ˜50-80 meters below the seafloor. The typical presentation of the seismic data in this case is a seismic section consisting of a large number of adjacent seismic traces, i.e., time series, for which amplitude is plotted versus two-way traveltime. Apart from this nearly exclusive application, seismic profiling has a potential of being used for the characterisation of the seabed sediments and the layered subbottom structure. The translation of traveltime into depth and, therefore, layer thickness, however, requires the knowledge of the acoustic properties—sound velocity and attenuation—of the sediments. While attenuation can be determined from traveltime and frequency content of the seismic pulse and the reflections, sound velocity can not uniquely be determined from single-channel seismograms without further knowledge. This further knowledge can be provided by a physically sound model conception that describes mathematically the quantitative relationships between acoustic properties and material properties of the seabed sediments.
1.2. Existing Theory and Previous Approaches
The most comprehensive existing mathematical/physical model to calculate the acoustic properties of a porous medium from the geometrical and elastic properties of its constituents and its structure, the frame, was formulated by Biot (1956a, b) (Biot, M. A. (1956a) “Theory of propagation of elastic waves in a fluid saturated porous solid, I. lower frequency range,” J. Acoust. Soc. Am., 28, 168-178) and (Biot, M. A. (1956b) “Theory of propagation of elastic waves in a fluid saturated porous solid, II. Higher frequency range,” J. Acoust. Soc. Am., 28, 179-191). This theory allows the calculation of velocity and attenuation of compressional- and shear waves of a fluid-filled porous solid from a set of sediment physical parameters:    Solid phase bulk density, ρr,    Solid phase bulk modulus, Kr,    Pore fluid density, ρf,    Pore fluid bulk modulus, Kf,    Pore fluid viscosity, η,    Frame bulk modulus, Kb,    Frame shear modulus, Gb,    Porosity, φ,    Permeability, κ,    Pore size, d,    Tortuosity, a.
In the frequency domain, velocity, VP, and attenuation, QP−1, of the compressional waves are given by
                                          V            P                    =                      ω                          ℜ              ⁡                              (                k                )                                                    ,                              1                          Q              P                                =                                    2              ⁢                              𝒥                ⁡                                  (                  k                  )                                            ⁢                              V                P                                                    π              ⁢                                                          ⁢              ω                                      ,                            (        1        )            where ω is angular frequency, (k) and ℑ(k) are real and imaginary parts of the complex wave number k which is calculated from the characteristic equation
                                                                                                                                                Hk                      2                                        -                                          ρω                      2                                                                                                                                                          ρ                        f                                            ⁢                                              ω                        2                                                              -                                          Ck                      2                                                                                                                                                              Ck                      2                                        -                                                                  ρ                        f                                            ⁢                                              ω                        2                                                                                                                                                        m                      ⁢                                                                                          ⁢                                              ω                        2                                                              -                                          Mk                      2                                        -                                                                  ⅈω                        ⁢                                                                                                  ⁢                        F                        ⁢                                                                                                  ⁢                        η                                            κ                                                                                                                =          0                ,                            (        2        )            where m=aρƒ/φ, ρ is the density of the saturated porous medium, F is a correction factor for flow at high frequencies, i is the imaginary unit, and
                              H          =                                                                      (                                                            K                      r                                        -                                          K                      b                                                        )                                2                                            D                -                                  K                  b                                                      +                          K              b                        +                                          4                3                            ⁢                              G                b                                                    ,                                  ⁢                  C          =                                                    K                r                            ⁡                              (                                                      K                    r                                    -                                      K                    b                                                  )                                                    D              -                              K                b                                                    ,                                  ⁢                  M          =                                    K              r              2                                      D              -                              K                b                                                    ,                                  ⁢                  D          =                                                    K                r                            ⁡                              (                                  1                  +                                      ϕ                    ⁡                                          (                                                                                                    K                            r                                                                                K                            f                                                                          -                        1                                            )                                                                      )                                      .                                              (        3        )            
This model, which in its original formulation is applicable to solid porous media only, has been adapted to unconsolidated marine sediments (Stoll, R. D. (1974) “Acoustic waves in saturated sediments,” in: Physics of sound in marine sediments, L. Hampton (ed.), New York: Plenum Press) by introducing the concept of an inelastic frame to replace the solid elastic frame. This inelastic frame is supposed to account for the fact that the solid phase in such an unconsolidated material occurs in a loosely-packed granular aggregate, a state that (1) shows a greatly reduced rigidity in contrast to the frame of a solid material and (2) absorbs energy of a passing sound wave. In this model, known as the Biot-Stoll model, the inelastic sediment frame is accounted for by constant complex bulk and shear frame moduli, Kb and Gb, respectively, that are assumed to describe a postulated grain-to-grain friction occurring at the grain contacts when an acoustic wave passes through the sediment. The two corresponding additional input parameters introduced by the Biot-Stoll model are the imaginary parts of the frame bulk and shear moduli. These are assumed according to empirical data (e.g., Hamilton, E. L. (1972) “Compressional wave attenuation in marine sediments,” Geophysics, 37, 620-646).
This concept of an inelastic sediment frame is physically problematic, because with the postulation of grain-to-grain friction, a non-linear component is added to the linear original Biot theory. This leads to a dependence on the magnitude of the particle displacements (Winkler, K., Nur, A., and Gladwin, M. (1978) “Friction and seismic attenuation in rocks,” Nature, 277, 528-531) of the wave attenuation which is not accounted for in the Biot-Stoll model. Moreover, the Biot-Stoll model does not allow for these complex bulk and shear frame moduli to be calculated from material properties but adjusts these moduli to fit literature data on sound attenuation in coarse-grained unconsolidated marine sediments such as sands. In addition, the Biot-Stoll model relies on empirical values for sound attenuation and lacks, therefore, mathematical rigour in the prediction of the acoustic properties of an arbitrary individual marine sediment. For fine-grained unconsolidated marine sediments such as clay, a further modification Stoll (Stoll, R. D. (1989) “Sediment Acoustics,” in: Bhattacharji, S, Friedman, G. M., Neugebauer, H. J., and Seilacher, A. (Eds.) Lecture Notes in Earth Sciences, Vol. 26, New York: Springer) to accommodate a fit with data introduces a viscoelastic component to the constant complex frame moduli. In spite of its partly empirical and heuristic nature, the Biot-Stoll model is widely used in modeling unconsolidated marine sediments (e.g., Turgut, A., and Yamamoto, T. (1988) “Synthetic seismograms for marine sediments and determination of porosity and permeability,” Geophysics, 53, 1056-1067); Holland, C. W., and Brunson, B. A. (1988) “The Biot-Stoll model: and experimental assessment,” J. Acoust. Soc. Am., 84, 1437-1443; and Turgut, A. (2000) “Approximate expressions for viscous attenuation in marine sediments: Relating Biot's “critical” and “peak” frequencies,” J. Acoust. Soc. Am., 108, 513-518). U.S. Pat. No. 5,815,465 (Sep. 29, 1998, Method and apparatus of classifying marine sediment (Altan Turgut) discloses a method to determine sediment parameters from acoustic measurements, in which Biot's equations are used in connection with constant complex elastic moduli of the sediment frame. This is an indication that actually the Biot-Stoll model is used therein, because in the original Biot theory the elastic moduli are real parameters.